This article studies the observer-based output feedback control problem for a class of cyber-physical systems with periodic denial-of-service (DoS) attacks, where the attacks coexist both in the measurement and control channels in the network scenario. The periodic DoS attacks are characterized by a cyclic dwell-time switching strategy, such that the resulting augmented system can be converted into a class of discrete-time cyclic dwell-time switched systems including a stable subsystem and an unstable subsystem. By means of a cyclic piecewise linear Lyapunov function approach, the exponential stability and <inline-formula><tex-math notation="LaTeX">l_2</tex-math></inline-formula>-gain analysis, and observer-based controller design are carried out for the augmented discrete-time cyclic switched system. Then, the desired observer and controller gains in piecewise linear form are determined simultaneously so as to ensure that the resulting closed-loop system is exponentially stable with a prescribed <inline-formula><tex-math notation="LaTeX">\mathcal {H}_{\infty }</tex-math></inline-formula> performance index. Finally, a practical application of unmanned ground vehicles under periodic DoS attacks is provided to verify the effectiveness of the developed control approach.